Consider a circular coil of wire carrying constant current I, forming a magnetic dipole. The magneti

Q: Why is the total flux through the infinite plane zero?

Because magnetic field lines always form closed loops. Every field line that comes out of the plane through the coil must eventually curve back and re-enter the plane to complete its loop. Thus, the net flux (outward + inward) is exactly zero.

Q: Does this logic apply to electric fields too?

No! Electric field lines can start on positive charges and end on negative charges. If you had a ring of positive charge, the electric field lines would go out to infinity and not necessarily loop back, so the net electric flux wouldn't be zero.

Q: What if the plane wasn't infinite, but just a large finite square?

If the plane were finite, some magnetic field lines would loop around the outer edges of the plane instead of piercing through it. In that case, the magnitude of $\phi_i$ would be strictly less than $\phi_o$.

Q: Why is there a negative sign in the answer?

Flux is a signed scalar. The negative sign indicates direction. If we define the flux coming OUT of the coil as positive, then the flux going INTO the plane must be negative. Hence, $\phi_i = -\phi_o$.

Q: Is the magnetic field uniform across the infinite plane?

Not at all! The magnetic field is strongest near the wire of the coil and gets rapidly weaker as you move further away (falling off as $1/r^3$ for a dipole). However, even though the field is highly non-uniform, the integral of the flux over the entire infinite area perfectly balances out.

Question

Consider a circular coil of wire carrying constant current $I$, forming a magnetic dipole. The magnetic flux through an infinite plane that contains the circular coil and excluding the circular coil area is given by $\phi_i$. The magnetic flux through the area of the circular coil area is given by $\phi_o$. Which of the following option is correct ?

Options:

(A) $\phi_i > \phi_o$

(B) $\phi_i < \phi_o$

(C) $\phi_i = \phi_o$

(D) $\phi_i = -\phi_o$

Sigma Percentile · Accepted Answer

### The Elegance of Closed Magnetic Loops

Imagine a massive, infinite plane stretching out endlessly in all directions. Right in the middle of this plane, we place a circular coil carrying a steady current $I$. This simple setup is a beautiful playground for one of the most fundamental laws of physics: **Gauss's Law for Magnetism**.

### The Nature of Magnetic Field Lines

Unlike electric charges, which can exist as isolated positive or negative point charges (monopoles), magnetic poles always come in pairs. You can never isolate a North pole from a South pole. Mathematically, this is expressed by Gauss's Law for Magnetism:

$$\oint \mathbf{B} \cdot d\mathbf{A} = 0$$

This equation tells us something profound: **magnetic field lines always form continuous, closed loops**. They have no starting point and no ending point.

### Visualizing the Flux

Let's apply this to our circular coil. The current $I$ creates a magnetic field, turning the coil into a magnetic dipole. Field lines emerge from one face of the coil (let's call it the North face) and must eventually loop back around to enter the other face (the South face).

Because the plane containing the coil is infinite, every single magnetic field line that shoots up out of the coil's area **must** pierce back down through the plane somewhere outside the coil to complete its journey. There is simply nowhere else for the field lines to go!

### The Mathematical Conclusion

Let the magnetic flux emerging from the circular coil area be $\phi_o$. 
Let the magnetic flux returning through the infinite plane (excluding the coil area) be $\phi_i$.

If we consider the entire infinite plane as a single boundary, the net flux through it must be zero because whatever comes out must go back in. Therefore, the sum of the outward flux and the inward flux is zero:

$$\phi_o + \phi_i = 0$$

By simply rearranging this equation, we arrive at our final, elegant answer:

$$\phi_i = -\phi_o$$

The negative sign perfectly captures the physical reality: the flux through the outer plane is exactly equal in magnitude but opposite in direction to the flux through the coil itself. It is a beautiful testament to the conservation of magnetic flux!

Select Answer:

Visualized Solution

$Visualizing the Setup$

$Gauss’s Law for Magnetism$

$Analyzing the Flux$

$Net Flux Equation$

$Final Conclusion$

$The Way Forward$

The Sigma Insight: Faraday's Laws of Electromagnetic Induction

The Elegance of Closed Magnetic Loops

The Nature of Magnetic Field Lines

Visualizing the Flux

The Mathematical Conclusion

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